Adjustment process-based approach for computing a Nash-Cournot equilibrium

This paper is focused on deriving properties of adjustment process-based algorithms for computing a Nash-Cournot equilibrium point. Two adjustment processes are considered: sequential and simultaneous. The corresponding numerical procedures are closely related to the Gauss-Seidel and Jacobi methods, respectively, for solving nonlinear systems. The present analysis shows that under quite general assumptions, both processes lead to closed algorithms. The paper also shows that in case of a linear demand function (which implies the absence of the diagonal dominance of the Jacobian matrix of marginal profiles), the sequential adjustment process-based algorithm converges to a Nash-Cournot equilibrium point. For this purpose the paper employs matrix splitting-based algorithms for solving a nonlinear complementarity problem, and an interesting equivalence between a Nash-Cournot equilibrium point and a certain n-dimensional optimization problem. This equivalence allows us also to gain more insight into why the simultaneous adjustment process fails to converge in this case.